Introduction

In Chapter 3 we formulated conditions describing when yielding may or may not occur. In this chapter we begin to explore what may happen if the stress point arrives at the yield surface. We intuitively expect that yielding will be accompanied by some form of increased deformation, over and above the elastic deformation that has gone on while the stress point has been inside the yield surface. We expect plastic behaviour to be softer than elastic behaviour, with the result that strains will accumulate more quickly. The term plastic flow is used to describe the deformation following yield.

One of the main differences between plastic response and elastic response is that plastic flow will be irreversible. While the material is elastic we can increase the stress with a consequent increase in strain, and then completely recover those strains by simply returning the stress state to its initial value. If yield occurs this will not be possible. Plastic deformation will not be recoverable from simple unloading. If we do reduce the stress to its initial value we will recover whatever elastic strain that has occurred in getting to the yield state, but the plastic strain will be locked within the body.

In order to describe plastic flow we might attempt to derive a constitutive relationship linking plastic strain to the current stress state. But this will immediately lead to difficulties owing precisely to the irreversibility mentioned above.

Plasticity and Geomechanics follows on from our earlier book Elasticity and Geomechanics. Like the earlier book, this one is very much a textbook rather than a treatise or reference book. It has grown from lecture notes and is written with students firmly in mind. Hopefully it will provide an easy, accessible introduction to a subject which, while being widely used in engineering practice, is often difficult for students to assimilate. The plasticity of metals is itself a subject of some complexity. When, instead of metals, the material we are concerned with is either soil or rock, the level of complexity is increased significantly. We have attempted here to untangle the ideas and concepts, and to lay out as clear a picture as possible of a subject area that is still in a state of development and discovery.

The book is organised as follows. Chapters 1 and 2 review some of the basic elements of stress and strain as well as the fundamentals of elasticity. Chapter 2 also presents a general discussion of inelastic response in soil, emphasising the defining characteristics of yield under isotropic compression and dilatancy as a result of shearing. Chapters 3 and 4 set out the fundamental ideas of yield surface and flow rules. The geometry of principal stress space is developed in detail. Yield loci for metals, for Coulomb materials and for some modifications of Coulomb materials are all presented. The Cam Clay and Modified Cam Clay surfaces are summarised.

The concept of the uniqueness of a solution is an essential requirement to the well-posed nature of a boundary value problem. A uniqueness theorem assures us that there is only one solution possible for the governing set of equations subject to appropriate boundary conditions. In EG we have discussed a uniqueness theorem in the context of the linear theory of elasticity. With linear theories in mechanics and physics, the development of a proof of uniqueness of solutions to boundary value problems and initial boundary value problems is well established. Comprehensive discussions of these topics are given in many texts on mathematical physics and on the theory of partial differential equations and also discussed in recent volumes by Selvadurai (2000a,b). The question that arises in the context of plasticity focuses on the development of a uniqueness theorem for what is basically a non-linear problem. This is not a straightforward issue, even with regard to certain situations involving non-linear behaviour of linear elastic materials. Examples that illustrate the concept of non-uniqueness of elasticity solutions can be readily found in problems dealing with elastic buckling of structural elements such as beam-columns and shallow shells under lateral loads. In these categories of problem the structure can exhibit multiple equilibrium states corresponding to the same level of loading. The purpose of the discussion given below is then to address the basic question of what constraints should be imposed, specifically regarding plastic stress–strain relations, in order that the solution to a particular boundary value problem is unique.

The formulation of any specific boundary value problem in geomechanics is greatly facilitated firstly by considering the specific attributes as they pertain to the geometry of the domain of interest. Other aspects of the formulation and solution can also include a consideration of features such as material symmetry and other geometric features of the loading and boundaries of the domain. For example, a two-dimensional plane strain problem involving the surface loading of a halfspace region by a concentrated line load (Figure A.1) is most conveniently formulated with reference to a plane polar coordinate system, whereas the plane strain problem involving surface loading by a distributed loading (Figure A.2) is formulated most conveniently in reference to a Cartesian coordinate system.

Also, referring to Figure A.3, the axisymmetric surface loading of a halfspace region by a concentrated load is most conveniently described in relation to a system of spherical polar coordinates, whereas the axisymmetric surface loading of a halfspace region is best formulated in relation to a system of cylindrical polar coordinates (Figure A.4).

While in the examples cited just previously, the choice of the coordinate system is largely dictated by the mode of loading, there are other situations where the geometrical boundaries of the domain of interest have a decided influence on the choice of the coordinate system.

Introduction

Geotechnical engineers have made good use of the theory of elasticity for a number of decades. It became clear near the end of the nineteenth century that a variety of problems involving an elastic halfspace could be solved using techniques developed by the French mathematician Joseph Boussinesq. Boussinesq solved the problem of a point load resting on the surface of a homogeneous isotropic linearly elastic halfspace. He also developed the solution for a rigid circular footing resting on the halfspace surface. His work inspired others to investigate related problems with the result that by the middle of the twentieth century a wide range of problems involving both homogeneous and layered halfspaces with isotropic and anisotropic elastic materials had been solved for a variety of loading conditions. Solutions continue to appear in the geotechnical literature as well as in other disciplines. There are also coupled solutions in which porous materials saturated with pore fluid are modelled incorporating both elastic deformation and pore fluid flow.

In this chapter we will outline the basic elements of behaviour of elastic materials. The stress–strain relations for isotropic materials are given in a variety of forms and relationships between the elastic constants are derived. We will note the bounds imposed on the elastic constants by thermodynamic requirements and we discuss some special classes of problems such as plane strain problems and problems involving incompressible materials. Much of this material is also presented in EG, often in more detail.

Introduction

How a material responds to load is an everyday concern for civil engineers. As an example we can consider a beam that forms some part of a structure. When loads are applied to the structure the beam experiences deflections. If the loads are continuously increased the beam will experience progressively increasing deflections and ultimately the beam will fail. If the applied loads are small in comparison with the load at failure then the response of the beam may be proportional, i.e. a small change in load will result in a correspondingly small change in deflection. This proportional behaviour will not continue if the load approaches the failure value. At that point a small increase in load will result in a very large increase in deflection. We say the beam has failed. The mode of failure will depend on the material from which the beam is made. A steel beam will bend continuously and the steel itself will appear to flow much like a highly viscous material. A concrete beam will experience cracking at critical locations as the brittle cement paste fractures. Flow and fracture are the two failure modes we find in all materials of interest in civil engineering. Generally speaking, the job of the civil engineer is threefold: first to calculate the expected deflection of the beam when the loads are small; second to estimate the critical load at which failure is incipient; and third to predict how the beam may respond under failure conditions.

The graphical construction for the representation of the state of stress at a point within a continuum region is generally attributed to the German engineer Otto Christian Mohr. Although the use of graphical techniques in structural and solid mechanics has been an important area of activity both for engineering calculations and stress analysis, particularly in the eighteenth and nineteenth centuries (see, e.g., Todhunter and Pearson (1886, 1893) and Timoshenko (1953), the contributions of Karl Culmann and Otto Mohr to the development of this area are regarded as being particularly significant. Despite the passage of time these graphical constructions have continued to serve as efficient educational tools for the visualisation of difficult concepts related to the representation of three-dimensional states of stress, particularly in relation to the description of failure states in materials. The fact that the techniques developed in relation to the stress state at a point that can be represented in terms of a stress matrix of rank two or a second-order tensor implies that the procedures are equally applicable to the description of other properties and states in continua, which can be described in a similar manner. Examples include the description of moments of inertia of solids, flexural characteristics of plates and the hydraulic conductivity characteristics of porous media, etc. The purpose of this Appendix is to present a brief outline of the significant features of Mohr circles and to develop the basic equations applicable to the three-dimensional graphical representation of the stress state at a point.

Introduction

One of the most powerful aspects of the theory of plasticity lies in its ability to easily predict approximate values for the collapse load in a very wide range of applications. This comes about through two theorems called the upper bound theorem and the lower bound theorem. As their names imply, the theorems provide bounds, or limiting values, for the collapse load. Often any usage of the theorems is referred to as limit analysis.

The business of predicting collapse loads is totally concerned with finding the loads that will bring the structure or body to an imminent state of collapse. We are not concerned with what happens before or after in the sense of trying to analyse elastic strains or plastic flow. Also, we must not confuse the collapse load with the yield load. In some instances they will be the same and yield will immediately lead to collapse, but in other cases yield may happen well before collapse. As an example, yield precedes collapse by a significant margin in the shallow foundation problem where localised yielding may happen immediately near the edges of a rigid footing, well in advance of the collapse load. There are restrictions on the applicability of both theorems. A key factor in the development of limit theorems rests with the normality relationship between the yield surface and its associated plastic strain rate vector. For either rigid–perfectly plastic or elastic–perfectly plastic materials, the limit theorems can be proved rigorously (see Appendix H).